Positivity of the second exterior power of the tangent bundles

See [9, Lemma 3.3] and [9, comments after Remark 3.7].  

By construction of $\mathop{\rm RatCurves}\nolimits^{n}(X)$, one has a morphism from ${\mathcal{M}}$ to the Chow scheme of $1$-cycles on $X$: $h:{\mathcal{M}}\to{\rm Chow}_{1}(X)$. The Chow scheme ${\rm Chow}_{1}(X)$ is projective [27, I. Theorem 3.21.3] and $h:{\mathcal{M}}\to h({\mathcal{M}})$ is finite; then ${\mathcal{M}}$ is not closed in ${\rm Chow}_{1}(X)$, because ${\mathcal{M}}$ is not proper. Thus there exists a $1$-cycle $Z\in\overline{{\mathcal{M}}}\setminus{\mathcal{M}}$. By [27, II, Proposition 2.2], the $1$-cycle $Z$ can be written as $Z=\sum a_{i}Z_{i}$, where each $a_{i}$ is a positive integer and each $Z_{i}$ is a rational curve.  

This follows from $\mathop{\rm det}\nolimits\left(\bigwedge^{2}T_{X}\right)=\omega_{X}^{1-n}$, where $n=\dim X$.  

Denoting by $f:{\mathbb{P}}^{1}\to X$ the normalization of $C$, there exists integers $a_{i}$ such that

By the assumption $C$ is not free, $a_{n}$ is negative. Since we have the natural inclusion $T_{{\mathbb{P}}^{1}}\to f^{\ast}T_{X}$, we have $a_{1}\geq 2$. Meanwhile, the nefness of $\bigwedge^{2}T_{X}$ implies that $a_{n-1}+a_{n}\geq 0$. Hence $a_{n-1}\geq-a_{n}>0$. As a consequence, we have

We have an exact sequence

Applying [18, Chapter II, Exercise 5.16 (d)], we obtain the following exact sequences:

for some vector bundle $E$ on $X$. Since $\bigwedge^{2}T_{X}$ is nef, so is $\varphi^{\ast}(\bigwedge^{2}T_{Y})$ by Proposition 2.1 (1). Thus the first assertion follows from Proposition 2.1 (2). Restricting the above exact sequences (2) and (3) to the fiber $F$, one has the following exact sequences:

If $\dim Y=1$, then $\varphi^{\ast}\left(\bigwedge^{2}T_{Y}\right)|_{F}=0$; thus it follows from the exact sequence (5) that $E|_{F}\cong\bigwedge^{2}\left(T_{X}\right)|_{F}$ is nef. If $\dim Y>1$, then $\varphi^{\ast}\left(\bigwedge^{2}T_{Y}\right)|_{F}\cong{\mathcal{O}}_{F}^{\oplus\binom{\dim Y}{2}}$: thus it follows from the exact sequence (5) and Proposition 2.1 (5) that $E|_{F}$ is nef. As a consequence, in any case $E|_{F}$ is nef. Then the exact sequence (4) and Proposition 2.1 (1) concludes that $T_{F}$ is nef.  

Applying Proposition 3.3 (2) to projections $p_{1}:X\to Y$ and $p_{2}:X\to Z$, the tangent bundles $T_{Y}$ and $T_{Z}$ are nef. Hence $T_{X}=p_{1}^{\ast}T_{Y}\oplus p_{2}^{\ast}T_{Z}$ is also nef.  

By Yau’s Theorem [52, Theorem 1], $X$ admits a Kähler-Einstein metric. Then the result of Kobayashi [26, Section 5.8] and Lübke [34] shows that the tangent bundle $T_{X}$ is $H$-semistable (in the sense of Mumford-Takemoto) with respect to any ample divisor $H$ on $X$. On the other hand, since $\bigwedge^{2}T_{X}$ is numerically flat, the second Chern class

is numerically trivial. This yields that $c_{2}(X)\cdot H^{n-2}=0$. Applying [41, IV, Theorem 4.1], $T_{X}$ is nef. Thus our assertion follows from [15].  

By [6, Theorem], we may assume that $n\geq 4$. Let $\varphi:X\to{\mathbb{P}}^{1}$ be a $K_{X}$-negative contraction. We take a minimal section $\tilde{\ell}\subset X$ of $\varphi$ as in Definition 3.6 (In this case, $X_{\ell}=X$ if we put ${\mathbb{P}}^{1}=\ell$). Let us take a family of rational curves ${\mathcal{M}}\subset\mathop{\rm RatCurves}\nolimits^{n}(X)$ containing $[\tilde{\ell}]$; then we claim that the family ${\mathcal{M}}$ is unsplit. If ${\mathcal{M}}$ were not unsplit, by Lemma 2.7 we may find a rational $1$-cycle $Z=\sum a_{i}Z_{i}$ as a degeneration of ${\mathcal{M}}$-curves, where each $a_{i}$ is a positive integer and each $Z_{i}$ is a rational curve. Then there exists a section of $\varphi$ among $Z_{i}$’s. This contradicts to the minimality of $\tilde{\ell}$. Thus ${\mathcal{M}}$ is unsplit.

For a general point $x\in{\rm Locus}({\mathcal{M}})$, it follows from [27, II Corollary 4.21] that the restriction of $\varphi$ to ${\rm Locus}({\mathcal{M}}_{x})$ is a finite morphism onto ${\mathbb{P}}^{1}$. This implies that $\dim{\rm Locus}({\mathcal{M}}_{x})=1$. Meanwhile, it follows from Proposition 2.8 that $\dim{\rm Locus}({\mathcal{M}}_{x})\geq\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}-1$. If there exists a non-free ${\mathcal{M}}$-curve, then by Lemma 3.2, we have

This contradicts to our assumption $n\geq 4$. Thus any ${\mathcal{M}}$-curve is free. As a consequence, ${\mathcal{M}}$ is an unsplit covering family such that any ${\mathcal{M}}$-curve is free. Applying Theorem 2.11, there exists a geometric quotient $\psi:X\to Z$ for ${\mathcal{M}}$ and it is a smooth morphism. Since any ${\mathcal{M}}$-curve is a section of $\varphi:X\to{\mathbb{P}}^{1}$, we see that $\varphi\times\psi:X\to{\mathbb{P}}^{1}\times Z$ is bijective; then by Zariski’s main theorem, we see that $\varphi\times\psi:X\to{\mathbb{P}}^{1}\times Z$ is an isomorphism. The remaining part follows from Corollary 3.4.  

Let $X$ be a smooth projective variety with nef $\bigwedge^{2}T_{X}$. By Theorem 4.1, there exists a finite étale cover $f:\tilde{X}\to X$ such that $\tilde{X}\cong Y\times Z$ where $\omega_{Y}\cong{\mathcal{O}}_{Y}$ and the Albanese morphism $\alpha_{Z}:Z\to{\rm Alb}(Z)$ is a locally trivial fibration such that the fiber $F$ is rationally connected. If $Y$ and $Z$ are positive-dimensional, then it follows from Corollary 3.4 that $T_{X}$ is nef. Thus our assertion follows from Theorem 1.1. Hence we assume that one of varieties $Y$ and $Z$ is a point. If $\tilde{X}\cong Y$, then our assertion holds thanks to Lemma 3.5. Thus assume that $\tilde{X}\cong Z$. If $\dim{\rm Alb}(\tilde{X})=0$, then $\tilde{X}\cong F$ is rationally connected; thus so is $X$. Since any smooth projective rationally connected variety is simply connected (see [13, Corollary 4.18 (b)]), we have $X\cong\tilde{X}$. So consider the case $\dim{\rm Alb}(\tilde{X})>0$. In this case, by Proposition 3.3 (2), $F$ is a rationally connected variety with nef $T_{F}$. Applying [15, Proposition 3.10], we see that $F$ is a Fano variety.  

We employ the notation as in the statement of Theorem 4.4. By [13, Corollary 3.15], $Y$ is an elliptic curve or a projective line ${\mathbb{P}}^{1}$. If $Y$ is ${\mathbb{P}}^{1}$, then our assertion follows from Proposition 3.7. Thus we assume that $Y$ is an elliptic curve. In this case, $\varphi:X\to Y$ factors through the Albanese map $\alpha_{X}:X\to{\rm Alb}(X)$:

Since $\varphi$ is a $K_{X}$-negative contraction, the morphism ${\rm Alb}(X)\to Y$ should be an isomorphism which in turn implies $\varphi$ is the Albanese map. Then by applying [7, Theorem 1.2], $\varphi$ is smooth as desired.  

By Proposition 4.5, we may assume $\dim Y>1$. Since general fibers of $\varphi:X\to Y$ are smooth Fano varieties, there exists a dominating family ${\mathcal{M}}$ of $\mathop{\rm RatCurves}\nolimits^{n}(X)$ such that any ${\mathcal{M}}$-curve is contracted by $\varphi$. By replacing if necessary, we may assume the anticanonical degree of the family ${\mathcal{M}}$ is minimal among such families; then we claim that ${\mathcal{M}}$ is locally unsplit. To prove this, fix an ample divisor $H$ on $Y$. Then, for sufficiently large $m\gg 0$, $-K_{X}+m\varphi^{\ast}H$ is ample and ${\mathcal{M}}$ is a dominating family with minimal degree with respect to an ample divisor $-K_{X}+m\varphi^{\ast}H$, so that ${\mathcal{M}}$ is locally unsplit. Applying Mori’s bend and break lemma, one has $n+1\geq\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}\geq\ell(R)$. If moreover $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}=n+1$, then it follows from Proposition 2.8 that ${\rm Locus}({\mathcal{M}}_{x})=X$ for a general point $x\in X$. Applying [27, II Corollary 4.21], we see that $\rho_{X}=1$; this is a contradiction. Thus we obtain an inequality

Here we claim the following:

Assume the contrary, that is, $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}>\ell(R)$. Choosing a rational curve $C$ in $R$ such that $\ell(R)=-K_{X}\cdot C$, $C$ is not free. In fact, if $C$ were free, then we could find a dominating family ${\mathcal{M}}^{\prime}$ of $\mathop{\rm RatCurves}\nolimits^{n}(X)$ such that any ${\mathcal{M}}^{\prime}$-curve is contracted by $\varphi$. However it contradicts to the minimality of the anticanonical degree of ${\mathcal{M}}$. Thus by Lemma 3.2 we have $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}=n$. Applying Proposition 2.8, for a general point $x\in X$,

As a general fiber of $\varphi$ contains ${\rm Locus}({\mathcal{M}}_{x})$, the relative dimension of $\varphi$ is $n-1$ which in turn implies $\dim Y=1$. This contradicts to our assumption that $\dim Y>1$. As a consequence, we see that $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}=\ell(R)$.

One can show that the family ${\mathcal{M}}$ is unsplit. If not, by Lemma 2.7 we could find a rational $1$-cycle $Z=\sum_{i=1}^{s}a_{i}Z_{i}$ as a degeneration of ${\mathcal{M}}$-curves, where each $a_{i}$ is a positive integer and each $Z_{i}$ is a rational curve. Then it follows from the extremality of the ray $R$ that each $[Z_{i}]$ is contained in $R$. However this contradicts to the minimality of $C$. Thus the family ${\mathcal{M}}$ is an unsplit covering family. Being ${\mathcal{M}}$ unsplit, we may consider a rationally connected fibration $\varphi_{{\mathcal{M}}}:X{\cdots}\to Y^{\prime}$ with respect to ${\mathcal{M}}$. If any ${\mathcal{M}}$-curve is free, then, applying Theorem 2.11, we see that $\varphi_{{\mathcal{M}}}:X\to Y^{\prime}$ is the geometric quotient of $X$ for ${\mathcal{M}}$ and it is a smooth morphism. By construction, the quotient morphism $\varphi_{{\mathcal{M}}}$ is nothing but $\varphi$. Hence in this case our assertion holds. Thus it is enough to show the following:

To prove this, assume the contrary; there would exist a non-free ${\mathcal{M}}$-curve $C_{0}$. By Lemma 3.2, one has $\ell(R)=-K_{X}\cdot C_{0}\geq n-1$. Applying Proposition 2.8, for a general point $x\in X$,

Thus the relative dimension of $\varphi_{{\mathcal{M}}}$ is at least $n-2$. By [2, Theorem 1 and its proof, Theorem 2], we see that $\varphi_{{\mathcal{M}}}:X\to Y^{\prime}$ is the geometric quotient and it is equidimensional. Being $\varphi_{{\mathcal{M}}}:X\to Y^{\prime}$ a contraction of a geometric extremal ray $R_{{\mathcal{M}}}$, $\varphi_{{\mathcal{M}}}$ is nothing but $\varphi$. In particular, $\varphi$ is equidimensional and one has inequalities

If $n-\dim Y=\ell(R)-1$, then [19, Theorem 1.3] tells us that $\varphi:X\to Y$ is a projective bundle. This contradicts to the existence of a non-free rational curve $C_{0}$. Hence we obtain $\dim Y=1$; however this contradicts to our assumption that $\dim Y>1$.  

If $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}\leq n-2$, then our assertion follows from Lemma 2.7, Lemma 3.2 and Theorem 2.11; thus suppose $\mathop{\rm deg}\nolimits_{(-K_{X})}{\mathcal{M}}>n-2$. Then, for a general point $x\in X$, by Proposition 2.8 we have $\dim{\rm Locus}({\mathcal{M}}_{x})\geq n-2$. Applying [2, Theorem 1 and 2], we see that there exists a geometric quotient $\varphi:X\to Y$ for ${\mathcal{M}}$, which is a contraction of a $K_{X}$-negative extremal ray $R_{{\mathcal{M}}}$. The smoothness of $\varphi$ follows from Theorem 4.4.  

Let $E$ be an irreducible component of the $\varphi$-exceptional locus and $F$ an irreducible component of any nontrivial fiber of $\varphi$. Then Proposition 2.2 shows that

By Lemma 3.2, one has $\ell(R)\geq n-1$ which in turn implies $\dim E=\dim F=n-1$ and $\ell(R)=n-1$. This means that $\varphi$ is a divisorial contraction such that $\varphi(E)$ is a point. This argument tells us that any birational contraction of a $K_{X}$-negative extremal ray is a divisorial contraction which contracts its exceptional divisor to a point.

Let us remark that $X$ is uniruled. In fact one has an étale cover $\tilde{X}$ of $X$ as in Proposition 4.2. By the existence of a $K_{X}$-negative extremal ray, the Albanese map $\alpha_{X}:\tilde{X}\to{\rm Alb}(\tilde{X})$ is a nontrivial Fano fibration; thus $\tilde{X}$ is uniruled. This yields that $X$ is also uniruled. Hence there exists a free rational curve on $X$. Thus we may take a minimal rational component ${\mathcal{M}}\subset\mathop{\rm RatCurves}\nolimits^{n}(X)$. We claim that ${\mathcal{M}}$ is unsplit. To prove this, assume the contrary; if not, by Lemma 2.7, we might find a rational $1$-cycle $Z=\sum_{i=1}^{s}a_{i}Z_{i}$ as a degeneration of ${\mathcal{M}}$-curves, where each $a_{i}$ is a positive integer and each $Z_{i}$ is a rational curve. Since $Z_{i}$ is not free thanks to the minimality of the anticanonical degree of ${\mathcal{M}}$, Lemma 3.2 implies that $-K_{X}\cdot Z_{i}\geq n-1$. Thus one obtains

By the same way as in the proof of Theorem 4.4, Mori’s bend and break lemma and [27, II Corollary 4.21] yield that $-K_{X}\cdot Z$ is at most $n$. This contradicts to our assumption that $n\geq 3$.

Applying Corollary 4.9, we obtain a smooth geometric quotient $\psi:X\to Z$ for ${\mathcal{M}}$. Since any fiber of $\psi$ should has dimension at most one, $\psi$ is a ${\mathbb{P}}^{1}$-bundle. By [16, Proposition 2.2 and its proof], $\psi:X\cong{\mathbb{P}}({\mathcal{O}}_{Z}\oplus{\mathcal{O}}_{Z}(m))\to Z$ and $E$ is its section, where $Z$ is an $(n-1)$-dimensional Fano manifold of $\rho_{Z}=1$ and ${\mathcal{N}}_{E/X}\cong{\mathcal{O}}_{E}(m)~{}(m<0)$. For any rational curve $D\subset E\cong Z$, it is not free as a curve on $X$; thus by Lemma 3.2

By Theorem 2.5 (1), $E\cong Z$ is isomorphic to ${\mathbb{P}}^{n-1}$. Moreover taking a line on $E\cong{\mathbb{P}}^{n-1}$ as a rational curve $D$ as in the above inequality, we obtain $m=-1$. As a consequence, $X$ is isomorphic to ${\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{n-1}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1))$, that is, $X$ is isomorphic to the blow-up of the projective space ${\mathbb{P}}^{n}$ at a point as desired.  

To prove our assertion, let us assume the contrary; then there exists a $K_{X}$-negative smooth contraction $\varphi:X\to Y\cong{\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{m}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{m}}(-1))$. We denote by $\beta:Y\to{\mathbb{P}}^{m+1}$ the blow-up of ${\mathbb{P}}^{m+1}$ at a point and by $E$ its exceptional divisor, which is a section $E={\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{m}}(-1))$ of a ${\mathbb{P}}^{1}$-bundle ${\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{m}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{m}}(-1))\to{\mathbb{P}}^{m}$. Denoting by $\ell$ a line on $E\cong{\mathbb{P}}^{m}$, let us consider $\varphi_{\ell}:X_{\ell}\to\ell={\mathbb{P}}^{1}$ and its section $\tilde{\ell}\subset X_{\ell}$ as in Definition 3.6. Note that $X_{\ell}$ coincides with $\varphi^{-1}(\ell)$; thus $\tilde{\ell}=\tilde{\ell}_{X}$ under the notation as in Definition 3.6. We have an exact sequence of normal bundles:

By [18, II, Proposition 8.10], we obtain